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SageMath
E = EllipticCurve("h1")
E.isogeny_class()
Elliptic curves in class 7605.h
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 7605.h1 | 7605p7 | \([1, -1, 1, -197730032, 1070230016756]\) | \(242970740812818720001/24375\) | \(85769379174375\) | \([4]\) | \(516096\) | \(3.0212\) | |
| 7605.h2 | 7605p5 | \([1, -1, 1, -12358157, 16724576756]\) | \(59319456301170001/594140625\) | \(2090628617375390625\) | \([2, 2]\) | \(258048\) | \(2.6746\) | |
| 7605.h3 | 7605p8 | \([1, -1, 1, -12061562, 17565245624]\) | \(-55150149867714721/5950927734375\) | \(-20939789837493896484375\) | \([2]\) | \(516096\) | \(3.0212\) | |
| 7605.h4 | 7605p3 | \([1, -1, 1, -790952, 248249954]\) | \(15551989015681/1445900625\) | \(5087753803244750625\) | \([2, 2]\) | \(129024\) | \(2.3281\) | |
| 7605.h5 | 7605p2 | \([1, -1, 1, -174947, -23777854]\) | \(168288035761/27720225\) | \(97540368772266225\) | \([2, 2]\) | \(64512\) | \(1.9815\) | |
| 7605.h6 | 7605p1 | \([1, -1, 1, -167342, -26305756]\) | \(147281603041/5265\) | \(18526185901665\) | \([2]\) | \(32256\) | \(1.6349\) | \(\Gamma_0(N)\)-optimal |
| 7605.h7 | 7605p4 | \([1, -1, 1, 319378, -134111194]\) | \(1023887723039/2798036865\) | \(-9845574761766749265\) | \([2]\) | \(129024\) | \(2.3281\) | |
| 7605.h8 | 7605p6 | \([1, -1, 1, 920173, 1174310804]\) | \(24487529386319/183539412225\) | \(-645828161664325878225\) | \([2]\) | \(258048\) | \(2.6746\) |
Rank
sage: E.rank()
The elliptic curves in class 7605.h have rank \(1\).
Complex multiplication
The elliptic curves in class 7605.h do not have complex multiplication.Modular form 7605.2.a.h
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 16 & 16 & 8 \\ 2 & 1 & 2 & 2 & 4 & 8 & 8 & 4 \\ 4 & 2 & 1 & 4 & 8 & 16 & 16 & 8 \\ 4 & 2 & 4 & 1 & 2 & 4 & 4 & 2 \\ 8 & 4 & 8 & 2 & 1 & 2 & 2 & 4 \\ 16 & 8 & 16 & 4 & 2 & 1 & 4 & 8 \\ 16 & 8 & 16 & 4 & 2 & 4 & 1 & 8 \\ 8 & 4 & 8 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
